Problem: Find the sum of the first $45$ terms in this geometric series: $-0.5 + 1.5 -4.5...$ Choose 1 answer: Choose 1 answer: (Choice A) A $-7.39\cdot10^{20}$ (Choice B) B $-4.92\cdot10^{20}$ (Choice C) C $ -3.69 \cdot 10^{20} $ (Choice D) D $1.23\cdot10^{20}$
Solution: Getting started We're dealing with a geometric series because each term is multiplied by $-3$ to get the next term. We need a formula to compute the sum of the terms. Formula for geometric series The sum $S_n$ of a finite geometric series is $S_n = \dfrac{a_1(1-r^n)}{1-r}$ where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = {-0.5})$ and the number of terms $(n = {45})$ are given in the question. The common ratio $r$ is ${-3}$ because each term is multiplied by ${-3}$ to get the next term. [How did we find the common ratio r?] Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac{a_1(1-r^n)}{1-r} \\\\ S_{{45}}&=\dfrac{{-0.5}(1-\left({-3}\right)^{{45}})}{1-\left({-3}\right)} \\\\ S_{{45}}&=-\dfrac18\left(1-\left({-3}\right)^{{45}}\right)\\\\ S_{{{45}}} &\approx -3.69 \cdot 10^{20} \end{aligned}$ The answer $ -3.69 \cdot 10^{20} $